Question

a right triangle is shown below. which equation represents a trigonometric ratio for the triangle? sin x = —— sin y = —— cos x = —— cos y = —— tan x = —— tan y = ——

Explanation:

Step1: Recall trigonometric ratio definitions

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

Step2: Find \(\sin x\)

For angle \(x\), the opposite side is \(TS\), and the hypotenuse is \(TR\). So \(\sin x=\frac{TS}{TR}\)

Step3: Find \(\sin y\)

For angle \(y\), the opposite side is \(RS\), and the hypotenuse is \(TR\). So \(\sin y = \frac{RS}{TR}\)

Step4: Find \(\cos x\)

For angle \(x\), the adjacent side is \(RS\), and the hypotenuse is \(TR\). So \(\cos x=\frac{RS}{TR}\)

Step5: Find \(\cos y\)

For angle \(y\), the adjacent side is \(TS\), and the hypotenuse is \(TR\). So \(\cos y=\frac{TS}{TR}\)

Step6: Find \(\tan x\)

For angle \(x\), the opposite side is \(TS\), and the adjacent side is \(RS\). So \(\tan x=\frac{TS}{RS}\)

Step7: Find \(\tan y\)

For angle \(y\), the opposite side is \(RS\), and the adjacent side is \(TS\). So \(\tan y=\frac{RS}{TS}\)

Answer:

\(\sin x=\frac{TS}{TR}\), \(\sin y=\frac{RS}{TR}\), \(\cos x=\frac{RS}{TR}\), \(\cos y=\frac{TS}{TR}\), \(\tan x=\frac{TS}{RS}\), \(\tan y=\frac{RS}{TS}\)